Higher order complexity Hugo Fre Mathieu Hoyrup CCA 2013 Hugo - PowerPoint PPT Presentation

Background & motivation Higher order strategies Higher order Turing machines Computable analysis Conclusion Higher order complexity Hugo Férée Mathieu Hoyrup CCA 2013 Hugo Férée Higher order complexity 1/12

Background & motivation Higher order strategies Higher order Turing machines Computable analysis Conclusion Type two Theory of Effectivity Computability • Represented spaces, admissibility (Weihrauch) • Extended admissibility, on QCB-spaces (Schröder) Complexity • Kawamura and Cook : Reg ⊆ Σ ∗ → Σ ∗ • Polynomial time complexity based on bff 2 • allows to define notions of complexity over non σ− compact spaces like C ([ 0 , 1 ] , R ) Hugo Férée Higher order complexity 1/12

Background & motivation Higher order strategies Higher order Turing machines Computable analysis Conclusion "Feasible" admissibility Definition (Polynomial reducibility) δ ≤ P δ ′ if δ = δ ′ ◦ f with f polynomial time computable Theorem (Kawamura & Cook) δ � is the "largest" representation of C ([ 0 , 1 ] , R ) making Eval : C ([ 0 , 1 ] , R ) → [ 0 , 1 ] → R polynomial time computable. → For which spaces can we do the same? Hugo Férée Higher order complexity 2/12

Background & motivation Higher order strategies Higher order Turing machines Computable analysis Conclusion First order representations are not sufficient Theorem Let X be a Polish space that is not σ -compact. Then there is no representation of C ( X , R ) making the time complexity of Eval X , R : C ( X , R ) × X → R well-defined. ( X = C ([ 0 , 1 ] , R ) for example) b Lemma There is no surjective partial continuous function φ : ( N → N ) → C ( N → N , N ) bounded by a total continuous function. Corollary "Higher order is required to define complexity-friendly representations." Hugo Férée Higher order complexity 3/12

Background & motivation Higher order strategies Higher order Turing machines Computable analysis Conclusion Higher order complexity Finite types: τ = N | τ 1 × . . . × τ n → N A notion of feasibility at all finite types: bff . Problem: Some intuitively feasible functionals are not in bff . Example 1 h i,n Γ : ( C ([ 0 , 1 ] , R ) → R ) × N → R � Γ( F , n ) = ( 1 + | F ( h i , n ) | ) 0 ≤ i ≤ 2 n ���� 2 − n 1 Hugo Férée Higher order complexity 4/12

Background & motivation Higher order strategies Higher order Turing machines Computable analysis Conclusion Higher order strategies Player � �� � g x f F (( N → N ) → N ) → N � �� � Opponent Moves: ? f or ! f ( v ) + justifications. Definition A strategy is a function which given a list of previous moves, outputs a valid move. Hugo Férée Higher order complexity 5/12

Background & motivation Higher order strategies Higher order Turing machines Computable analysis Conclusion Examples • x = 3 ? x ! x (3) • f ( x ) = 2 x + 1 ? x ! x ( n ) ? f ! f (2 n + 1) • F ( g ) = g ( λ x . x ) + 1 ? F ! g ( n ) ? g ! F ( n + 1) ? h ? x ! x ( n ) ! g ( m ) ! h ( n ) ! F ( m + 1) ? h ? x . . . Hugo Férée Higher order complexity 6/12

Background & motivation Higher order strategies Higher order Turing machines Computable analysis Conclusion Examples • x = 3 ? x ! x (3) • f ( x ) = 2 x + 1 ? x ! x ( n ) ? x ! x ( n ) ? x ! x ( n ) ? f ! f (2 n + 1) • F ( g ) = g ( λ x . x ) + 1 ? F ! g ( n ) ? g ! F ( n + 1) ? h ? x ! x ( n ) ! g ( m ) ! h ( n ) ! F ( m + 1) ? h ? x . . . Hugo Férée Higher order complexity 6/12

Background & motivation Higher order strategies Higher order Turing machines Computable analysis Conclusion Examples • x = 3 ? x ! x (3) • f ( x ) = 2 x + 1 ? x ! x ( n ) ? x ! x ( n ) ? x ! x ( n ) ? f ! f (2 n + 1) • F ( g ) = g ( λ x . x ) + 1 ? F ! g ( n ) ? g ! F ( n + 1) H ? h ? x ! x ( n ) ! g ( m ) ! h ( n ) ! F ( m + 1) ? h ? x . . . Hugo Férée Higher order complexity 6/12

Background & motivation Higher order strategies Higher order Turing machines Computable analysis Conclusion Size of a strategy Definition S s ( b 1 , . . . , b n ) = | H ( s , s 1 , . . . s n ) | max ( s 1 ,... s n ) ∈ K b 1 ×···× K bn K b = { s ′ | S s ′ � b } Example • n ∈ N has a strategy of size O ( log 2 n ) . • f : N → N has a strategy of size | f | ( n ) = max | x |≤ n | f ( x ) | . • The size of a strategy for F : ( N → N ) → N is at least its modulus of continuity. Hugo Férée Higher order complexity 7/12

Background & motivation Higher order strategies Higher order Turing machines Computable analysis Conclusion Higher order Turing machines Definition (HOTM) A HOTM is a kind of oracle Turing machine which plays a game versus its oracles. Property A strategy is computable ⇐ ⇒ it is represented by a HOTM. Run-time of a HOTM: same as for an OTM. Hugo Férée Higher order complexity 8/12

Background & motivation Higher order strategies Higher order Turing machines Computable analysis Conclusion Polynomial time complexity Definition (Higher type polynomials ) HTP = simply-typed λ − calculus with + , ∗ : N × N → N . Property HTP of type 1 and 2 are respectively the usual polynomials and the second-order polynomials. Example The complexity of Γ is about F , n �→ F ( λ x . c ) × F ( λ y . P ( y , n )) Hugo Férée Higher order complexity 9/12

Background & motivation Higher order strategies Higher order Turing machines Computable analysis Conclusion Higher order representations Definition (Kleene-Kreisel Spaces) KKS = [ N , ⊆ , → , × ] Definition (Representation) A representation δ of a space X with a KKS A is a surjective function from A to X . Definition (Polynomial reduction) δ 1 ≤ P δ 2 if δ 1 = δ 2 ◦ F for some polynomial time computable F : A 1 → A 2 . Hugo Férée Higher order complexity 10/12

Background & motivation Higher order strategies Higher order Turing machines Computable analysis Conclusion Standard representation of C ( X , Y ) Definition δ C ( X , Y ) ( F ) = f whenever f ◦ δ X = δ Y ◦ F Property Eval : C ( X , Y ) × X → Y is polynomial-time computable w.r.t. (δ C ( X , Y ) , δ X , δ Y ) Theorem It is the largest representation making Eval polynomial. Hugo Férée Higher order complexity 11/12

Background & motivation Higher order strategies Higher order Turing machines Computable analysis Conclusion Conclusion • we have a definition of higher order complexity • new representation spaces • we need to understand the difference with bff • study the notion of admissibility of such representations (c.f. Schröder). Hugo Férée Higher order complexity 12/12